Palestinian militants had begun a "war" against Israel which they infiltrated by air, sea and land from the blockaded Gaza Strip on Saturday, Israeli officials said, a major escalation in the Israeli-Palestinian conflict.
Israeli medics reported 40 people killed.
"Since the morning hours, MDA teams provided medical care to hundreds of casualties, and pronounced 40 people dead," it said, while the Israeli health ministry confirmed that at least 779 people had been injured and taken to hospitals.
Prime Minister Benjamin Netanyahu said "We are at war" and vowed severe retaliation after ordering an extensive mobilisation of army reserves. "The enemy will pay an unprecedented price" for the surprise attack, he warned.
Iran-backed Hamas, which controls the Gaza Strip, released a video showing its fighters had captured three men dressed in civilian clothes and described as "enemy soldiers" in the video caption.
"We decided to put an end to all the crimes of the occupation (Israel). Their time for rampaging without being held accountable is over," said the Ezzedine al-Qassam Brigades, the Hamas armed wing.
"We announce Operation Al-Aqsa Flood and we fired, in the first strike of 20 minutes, more than 5,000 rockets."
Israel's army said its forces were fighting Palestinian militants on the ground in several locations near the Strip. It dubbed its operation "Swords of Iron".
Army spokesman Richard Hecht said the militants conducted a combined raid "which happened through paragliders, through the sea and through the ground".
He would not be drawn on reports that Israelis had been captured.
Unverified videos on social media showed the bodies of a number of people in military fatigues as well as dead motorists and passengers on a highway.
"Send help, please!" one woman sheltering with her two-year-old child pleaded as militants shot at her house and tried to break into their safe room, according to Israeli media.
Militant infiltration from Gaza, an impoverished enclave home to 2.3 million people, has been rare since Hamas took control in 2007, leading to Israel's crippling blockade. Gaza is sealed off from Israel by a militarised border barrier.
The rocket barrage from Gaza -- which Hecht said numbered at least 2,200 -- left cars burning beneath residential buildings in the Israeli city of Ashkelon, about 10 kilometres (six miles) north of Gaza.
'Gates of hell'
The attack occurred on Shabbat during a Jewish holiday.
Netanyahu said in a statement that the Hamas attack would be riposted at "a magnitude that the enemy has not known".
Israeli Major General Ghasan Alyan warned Hamas had "opened the gates of hell" and would "pay for its deeds".
AFP journalists said Israel's military began air strikes on Gaza after rockets began streaming across the sky from inside the territory beginning at 6:30 am (0330 GMT).
The military said dozens of its fighter jets "are currently striking a number of targets belonging to the Hamas terrorist organisation in the Gaza Strip."
An AFP journalist saw armed Palestinians gathered around an Israeli tank, which was partially in flames after they crossed the border fence from Khan Yunis in Gaza.
Another AFP journalist saw Palestinians returning to Gaza City driving a seized Israeli Humvee.
Air raid sirens wailed across southern and central Israel, as well as an unusual number of times in Jerusalem, where AFP journalists heard multiple rockets being intercepted by Israeli air defence systems.
The army urged people to stay near bomb shelters.
Hundreds of Gazans flee
Israeli police set up roadblocks to check motorists on the highway between Tel Aviv and Jerusalem, an AFP journalist said.
Hundreds of residents fled their homes in Gaza to move away from the border with Israel, mostly in the territory's northeast, an AFP correspondent said, adding the men, women and children carried blankets and food.
In the Israeli commercial centre of Tel Aviv, residents were seen boarding a bus to seek safety in a hotel.
An AFP photographer in the city saw a gaping hole in a building, with residents gathered outside.
In Israeli-annexed east Jerusalem, some Palestinian residents cheered and blew their car horns as sirens blared.
Israel's Magen David Adom emergency medical services said that, by early afternoon, its teams "have pronounced 40 victims deceased" from gunshots, while hundreds more were wounded.
Among the dead was the president of a regional council for Israeli communities northeast of Gaza. The council said its president was killed in an exchange of fire with Gaza attackers.
Hamas calls to 'join battle'
Hamas called on "the resistance fighters in the West Bank" as well as "our Arab and Islamic nations" to join the battle, in a statement posted on Telegram.
An adviser to Iran's supreme leader said he was "proud" of the Hamas action.
Lebanese militant group Hezbollah, which fought a devastating war against Israel in 2006, hailed the Palestinians' "heroic operation on a grand scale".
The Gaza fighting erupted as Washington attempted to mediate a landmark deal that would see Israel and Middle East powerhouse Saudi Arabia establish diplomatic ties.
Western capitals roundly denounced the Palestinian attacks on Israel.
European Commission President Ursula von der Leyen called it "terrorism in its most despicable form," while her foreign policy chief expressed "solidarity with Israel".
The EU, US and Israel consider Hamas to be a terrorist group.
United Nations Middle East peace envoy Tor Wennesland said the assault against Israeli communities and civilians has led to "a dangerous precipice and I appeal to all to pull back from the brink".
The violence follows heightened tensions in September when Israel closed the border to Gazan workers for two weeks.
That shutdown came as Palestinian demonstrators along the border burned tyres and threw rocks and petrol bombs at Israeli troops, who responded with tear gas and live bullets.
In May, an exchange of Israeli air strikes and Gaza rocket fire killed 34 Palestinians and one Israeli.
Violence between Israel and the Palestinians has been surging since early last year.
This year's rise in unrest came against the backdrop of divisive judicial reforms introduced by the hard-right government of Netanyahu, who is on trial for corruption charges he denies.
Before Saturday's violence at least 247 Palestinians, 32 Israelis and two foreigners had been killed in the conflict, including combatants and civilians on both sides, according to Israeli and Palestinian officials.
The vast majority of fatalities have occurred in the West Bank, which has been occupied by Israel since the 1967 Arab-Israeli War.
